We have simplified the rational functions \(r^{(\ell ),i,c}\)  appearing in the amplitudes by expressing them as linear combinations of a much smaller set of rational functions. Those have been further simplified with the help of the partial fractioning package MultivariateApart [1]}. In the following we will be referring to this much smaller set as independent rational functions. The number of (independent) rational functions for each colour/charge structure can be found in table REF . The appearance of these simpler rational structures has already been discussed in the literature [2]}, [3]}, [4]}, [5]}, [6]} and they have been utilized for expressing the amplitudes in a more compact from. On the other hand, it seems to us that the possible true structure behind these independent rational functions has not yet been fully explored and we hope to return to this in a future publication.
